The complete expansion can be found in Appendix . The fact that we keep \(m_{\rm eff}=m+g\Phi \)  of adiabatic order zero but the derivative of the field \(\dot{\Phi }\)  of adiabatic order is not arbitrary. If we were to choose \(\Phi \)  of adiabatic order zero such that \(\dot{\Phi }\)  is of adiabatic order one it would be inconsistent with the subtraction terms computed for the two point function and the stress-energy tensor such that the renormalized observables would still contain divergences. This expansion is equivalent as the one obtained in [1]} for an scalar field and for more general spacetimes the R-summed solution via Schwinger-DeWitt expansion [2]}.
